Thermal Physics - Superconductor

Which of the following curves is characteristic of the specific heat C_v of a metal such as lead, tin, or aluminum in the temperature region where is becomes superconducting?

(GR9677 #95)

Solution:




Answer: E

Nuclear & Particle Physics – Scattering cross section

A beam of 10¹² protons per second is incident on a target containing 10²⁰ nuclei per square cm. At an angle of 10 degrees, there are 10² protons per second elastically scattered into a detector that subtends a solid angle of 10⁻⁴ steradians. What is the differential elastic scattering cross section, in units of sq cm per steradian?

A. 10²⁴
B. 10²⁵
C. 10²⁶
D. 10²⁷
E. 10²⁸

(GR9277 #95)

Solution:

Using dimension analysis:

the fractional scattering rate:



In units of sq cm per steradian:



Answer: C

Thermal Physics - Carnot Cycle

In the cycle shown above, KL and NM represent isotherms, while KN and LM represent reversible adiabats. A system is carried through the Carnot cycle KLMN, taking in heat Q₂ from the hot reservoir T₂ and releasing heat Q₁ to the cold reservoir T₁. All of the following statements are true, EXCEPT:

A. Q₁/T₁ = Q₂/T₂
B. The entropy of the hot reservoir decreases
C. The entropy of the system increases
D. The work done is equal to the net heat absorbed, Q₂ − Q₁
E. The efficiency of the cycle is independent of the working substance.

(GR8677 #95)

Solution:

Carnot engine efficiency:

where W = Q₂ − Q₁ → work done
(D) TRUE

The efficiency η does not depend on the working substance
(E) TRUE

W = Q₂ − Q₁
η = (Q₂ − Q₁)/Q₂ = 1 − (Q₁/Q₂) = 1 − (T₁/T₂)
Q₁/Q₂ = T₁/T₂
Q₁/T₁ = Q₂/T₂
(A) TRUE

The hot reservoir has decreasing entropy because it gets cooler as the cycle proceeds.
(B) TRUE

In order to approach the Carnot efficiency, the processes involved in the heat engine cycle must be reversible and involve no change in entropy.
(C) FALSE

Answer: C

Electromagnetism – Dielectric

An infinite slab of insulating material with dielectric constant K and permittivity ɛ = Kɛ₀ is placed in a uniform electric field of magnitude E₀. The field is perpendicular to the surface of the material. The magnitude of the electric field inside the material is 

A. E₀/K
B. E₀/Kɛ₀ 
C. E₀
D. Kɛ₀E₀
E. KE₀

(GR0177 #95)

Solution:

In vacuum, E₀ = σ/ɛ₀
In dielectric, E = σ/ɛ
where ɛ = Kɛ₀
Thus, E = σ/Kɛ₀ = E₀/K

Answer: A

Quantum Mechanics - Commutation Relations

Let be a quantum mechanical angular momentum operator. The commutator [, ] is equivalent to which of the following?

A. 0
B. iħ
C. iħ
D. −iħ
E. iħ

(GR0877 #95)

Solution: 

Commutator identities: [A, B] = − [B, A]

and, [B, AC] = A[B, C] + [B, A]C 

[, ] = − [, ]

A = 
B = 
C = 

[, ] = [, ] +  [, ]
= [, ] + 0
= [iħ] + 0

− [, ] = −iħ

Answer: D